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<div class="add">

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:31_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:31_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:31_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E3:31_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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